Space of Somewhere Differentiable Continuous Functions on Closed Interval is Meager in Space of Continuous Functions on Closed Interval

Theorem

Let $I = \closedint a b$.

Let $\map \CC I$ be the set of continuous functions on $I$.

Let $\map \DD I$ be the set of continuous functions on $I$ that are differentiable at a point.

Let $d$ be the metric induced by the supremum norm.

Then $\map \DD I$ is meager in $\struct {\map \CC I, d}$.

Corollary

there exists a function $f \in \map \CC I$ that is not differentiable anywhere.

Proof

Let:

$\ds A_{n, m} = \set {f \in \map \CC I: \exists x \in I: \forall t \in I: 0 < \size {t - x} < \frac 1 m \implies \size {\frac {\map f t - \map f x} {t - x} } \le n}$

and:

$\ds A = \bigcup_{\tuple {n, m} \mathop \in \N^2} A_{n, m}$

Lemma 1

$\map \DD I \subseteq A$

$\Box$

Lemma 2

for each $\tuple {n, m} \in \N^2$, $A_{n, m}$ is nowhere dense in $\struct {\map \CC I, d}$.

$\Box$

From Lemma 2:

$A$ is the countable union of nowhere dense sets.

So, by definition of a meager space:

$A$ is meager in $\struct {\map \CC I, d}$.

By Subset of Meager Set is Meager Set and Lemma 1:

$\map \DD I$ is meager in $\struct {\map \CC I, d}$.

$\blacksquare$